Representations of bounded distributive lattices as the continuous sections of Sheaves based on the Priestly and Zarski topologies

TL;DR

This paper develops a sheaf-theoretic representation for bounded distributive lattices with modalities, linking algebraic structures to topological sheaves, and explores applications in many-valued and modal logics.

Contribution

It introduces a novel sheaf-based representation theorem for bounded distributive lattices with modalities, expanding the algebraic-topological correspondence.

Findings

Representation via continuous sections of sheaves is established.

Applications to many-valued and modal logics are demonstrated.

The framework unifies algebraic and topological approaches in logic.

Abstract

Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor. We give applications to many-valued logics logics and various modifications of first order logic and multi-modal logic, set in an algebraic framework.